1calculating the domain of attraction: zubov's method and extensions · 2005. 12. 18. ·...

Calculating the domain of attraction:Zubov’s method and extensions

Fabio Camilli 1 Lars Grune 2 Fabian Wirth 3

1University of L’Aquila, Italy

2University of Bayreuth, Germany

3Hamilton Institute, NUI Maynooth, Ireland

SCP 2005, St. Petersburg, June 29 – July 1, 2005

Overview

Problem descriptionThe domain of attractionProperties of the domain of attraction

Zubov’s approachZubov’s equationSeries approximation

Robust domains of attractionProblem statementA robust version of Zubov’s theoremExamples

Asymptotic controllabilityProblem descriptionZubov’s theorem for control systemsExamples

Problem description Zubov’s approach Robust domains of attraction Asymptotic controllability

The domain of attraction

Consider a nonlinear system

x(t) = f (x(t)) , t ∈ R (1)

x(0) = x0 ∈ Rn ,

f Lipschitz continuous, f (0) = 0.Assume x∗ = 0 is asymptoticallystable.

x∗ = 0

The domain of attraction of 0 is defined by

A(0) := {x ∈ Rn | ϕ(t; x) → 0, as t →∞} .

Here ϕ(·; x) denotes the solution of (1).


The domain of attraction: Properties

A(0) has the following properties:

I A(0) is diffeomorphic to Rn,

I Conversely, any set, that is diffeomorphic to Rn, is the domainof attraction of some asymptotically stable fixed point,

I dynamic properties:A(0) and clA(0) are invariant under the flow,

There is little general information available on the domain ofattraction. On the other hand, in applications one is ofteninterested in knowing, what it looks like.Question:How to compute (approximations of) the domain of attraction?


Zubov’s result

V.I. Zubov:Methods of A.M. Lyapunov and their applicationNordhoff, Groningen, The Netherlands, 1964(Russian edition in 1957)


Zubov’s result

x = f (x) , t ∈ R (1)

TheoremA set A containing 0 in its interior is the domain of attraction of(1) if and only if there exist continuous functions V , h such that

I V (0) = h(0) = 0,0 < V (x) < 1 for x ∈ A \ {0}, h > 0 on Rn \ {0}

I for every γ2 > 0 there exist γ1 > 0, α1 > 0 such that

V (x) > γ1 , h(x) > α1 , if ‖x‖ ≥ γ2 ,

I V (xn) → 1 for xn → ∂A or ‖xn‖ → ∞,

I DV (x) · f (x) = −h(x)(1− V (x))√

1 + ‖f (x)‖2


Zubov’s result II

By proper choice of h the function V can be made as smooth as f0.In particular:

CorollaryIf f is continuously differentiable, then

DV (x) · f (x) = −h(x)(1− V (x))√

1 + ‖f (x)‖2 .

has at most one continuously differentiable solution satisfyingV (0) = 0.In order for the solution to exist it is sufficient that∫ ∞

0h(ϕ(t, x0))dt < ∞ .

for all x0 sufficiently small.


Zubov’s result III: Series approximation

DV (x) · f (x) = −h(x)(1− V (x))√

1 + ‖f (x)‖2 (Z ) .

If f is real-analytic, then h,V may also be chosen real analytic.Series approximations of V may be calculated by expanding:

V (x) = 0 + 0 + xTPx +O(3)

h(x) = 0 + 0 + xTQx +O(3)

f (x) = 0 + Ax +O(2)

Then the second order terms in (Z) yield

xTPAx = −xTQx ⇔ ATP + PA = −1

2Q .


Robust domains of attraction

Consider systems

x(t) = f (x(t), d(t)) , t ∈ R

with a perturbation term d . We are interested in robust stabilityproperties. In particular, the robust domain of attraction.

AD(0)


Robust domains of attraction

x(t) = f (x(t), d(t)) , t ∈ R

Assumptions:• f : Rn × D → Rn continuous, locally Lipschitz continuous in x ,uniformly in d• D ⊂ Rm compact, convex, d(t) ∈ D a.e.• f (0, d) = 0 for all d ∈ D• 0 is locally uniformly asymptotically stableNotation: D := {d : R → D ; d Lebesgue measurable}Question:Can the approach of Zubov be used to determine Lyapunovfunctions on the robust domain of attraction AD(0)?


A robust version of Zubov’s theorem

Theorem:[Zubov’s theorem for perturbed systems]Under suitable conditions on g there is a unique viscosity solutionof {

infd∈D

{−Dv(x)f (x , d)− (1− v(x))g(x , d)} = 0

v(0) = 0

The robust domain of attraction satisfies

AD(0) = v−1([0, 1)) .

This is a straightforward generalization of Zubov’s original equation

DV (x) · f (x) = −h(x)(1− V (x))√

1 + ‖f (x)‖2 .

for which we hadA = V−1([0, 1)) .


A robust version of Zubov’s theorem: Outline of proof

I choose g continuous, positive definite, such that∫ ∞

0g(φ(t, x0, d), d(t))dt < ∞

if and only if φ(t, x0, d) → 0.To be able to do this information on the local convergencerate near x∗ = 0 is needed.

I define v via the optimal control problem

v(x) = supd∈D

[1− exp

(∫ ∞

0g(φ(t, x0, d))dt

)].

I Use relation between Bellman principle and viscosity solutionsto show that v solves the robust Zubov equation.

I Prove uniqueness.


Example

x1 = −x1 + x2

x2 = 0.1x1 − 2x2 − x21 − 0.1x3

1

Fixed points:[0, 0], unstable[−2.5505,−2.5505], asymptotically stable[−7.4495,−7.4495], asymptotically stable.


Example

x1 = −x1 + x2

x2 = 0.1x1 − 2x2 − x21

− (0.1 + d(t)) x31

D = {0}


Example

x1 = −x1 + x2

x2 = 0.1x1 − 2x2 − x21

− (0.1 + d(t)) x31

D = [−0.02, 0.02]


A toy example

x1 = −x1 + d(t)x21

x2 = −x2 + d(t)x22

D = [−2, 2]


Another toy example

Khalil, Nonlinear Systems, 2nd ed., p. 190

x(t) = −2x(t) + x(t)y(t)

y(t) = −y(t) + (1 + d(t))x(t)y(t) .


Another toy example

x = −2x + xy

y = −y + (1 + d(t))xy .

D = [−1, 1]

In reality, the whole lower half plane is contained in the robustdomain of attraction. The discrepancy to the picture is caused bycalculating on a finite domain.


Asymptotic nullcontrollability

Consider the control system

x = f (x , u) ,

where f : Rn × U → Rn is sufficiently regular (to be made precise)and U ⊂ Rm is closed.Assume that f (0, 0) = 0.We are now interested in asymptotic nullcontrollability.The admissible controls are given by

U := {u : R+ → U | measurable and essentially bounded } .

The domain of asymptotic nullcontrollability is defined by

AU(0) := {x ∈ Rn | there exists u ∈ U with ϕ(t, x , u) → 0 for t →∞} .


Control Lyapunov functions

x = f (x , u) , f (0, 0) = 0 .

AU(0) := {x ∈ Rn | there exists u ∈ U with ϕ(t, x , u) → 0 for t →∞} .

It is known that asymptotic nullcontrollability is closely tied to theexistence of a control Lyapunov function:Consider a continuous, positive definite1, proper2 functionv : Rn → R+.v is called a control Lyapunov function on AU(0) if there is apositive definite function W such that v is a viscosity supersolutionon AU(0) of

maxu∈U,‖u‖≤κ(‖x‖)

−Dv(x)f (x , u) ≥ W (x) .

1v(x) ≥ 0 and v(x) = 0⇔ x = 02preimages of compact sets are compact


A Zubov type theorem

Theorem:[Zubov’s theorem for controlled systems]For suitable g there exists a unique viscosity v solution of{

supu∈U{−Dv(x)f (x , u)− (1− v(x))g(x , u)} = 0v(0) = 0

In particular, v is a control Lyapunov function on

AU(0) = v−1([0, 1)) .

Again this is a straightforward generalization of Zubov’s originalequation

Dv(x) · f (x) = −h(x)(1− v(x))√

1 + ‖f (x)‖2 .


Comments on proof:

Again proof relies on the analysis of an optimal control problem,namely

v(x) = infu∈U

[1− exp

(∫ ∞

0g(φ(t, x0, u))

)].

The choice of g is a bit more complicated this time, because wehave to deal with the unboundedness of u.Still explicit conditions, that are easily checkable are available.


Example

Consider the inverted pendulum

x1 = x2

x2 = − sin(x1 + π)− x2 + u


Example

x1 = x2

x2 = − sin(x1 + π)− x2 + u

U = [−0.7, 0.7]

-4 -3 -2 -1 0 1 2 3 4-5-4-3-2-1012345

00.10.20.30.40.50.60.70.80.9

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Example

x1 = x2

x2 = − sin(x1 + π)− x2 + u

U = [−1, 1]

-4 -3 -2 -1 0 1 2 3 4-5-4-3-2-1012345

00.10.20.30.40.50.60.70.80.9

1


Example

-4 -3 -2 -1 0 1 2 3 4-5-4-3-2-1012345

00.10.20.30.40.50.60.70.80.9

1


Numerical treatment

Consider the perturbed case:Zubov’s PDE has a singularity at 0.{

infd∈D

{−Dv(x)f (x , d)− (1− v(x))g(x , d)} = 0

v(0) = 0

To make the numerical treatment rigorous we consider the function

gε(x , d) = max{g(x , d), ε}

for fixed ε > 0 and the approximate equation{infd∈D

{−Dv(x)f (x , d)− g(x , d) + v(x)gε(x , d)} = 0

v(0) = 0


Numerical treatment II

Theorem Let v be the unique solution of Zubov’s equation. Thenfor each ε > 0 the approximate equation has a unique continuousviscosity solution vε with the following properties.

(i) vε(x) ≤ v(x) , ∀x ∈ Rn

(ii) vε → v uniformly in Rn as ε → 0

(iii) If ε < g0 then we have the characterization

AD(0) = {x ∈ Rn | vε(x) < 1}

(iv) If f (·, d) and g(·, d) are uniformly Lipschitz (uniformly in Dwith Lipschitz constants Lf and Lg ) and further technicalassumptions on g are satisfied then vε is uniformly Lipschitzon Rn.

1calculating the domain of attraction: zubov's method and extensions · 2005. 12. 18. ·...

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